Rule of Sarrus - Citizendia

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3x3 matrix. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n It is named after the French mathematician Pierre Frédéric Sarrus. Pierre Frédéric Sarrus (10 March 1798 Saint-Affrique - 20 November 1861 was a French Mathematician.

Sarrus rule: solid diagonals - dashed diagonals

Consider a 3x3 matrix $M=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$ , then its determinant can be computed by the following scheme:

Repeat the first 2 columns of the matrix behind the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields:

$M=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}$

A similar scheme based on diagonals works for 2x2 matrices: $M=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{21}a_{12}$

Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. In Algebra, the Leibniz formula expresses the Determinant of a square matrix A = (a_{ij}_{ij = 1 \dots n} in terms of permutations of the matrix'

References

Gerd Fischer: Analytische Geometrie. 4-te Auflage, Vieweg 1985, ISBN 3-528-37235-4, P. 145 (German)
Sarrus' rule at Planetmath

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